Pure Mathematics Practice Pack: Past Paper Style Questions
Pure Mathematics Practice Pack
Past paper style questions with full mark scheme, AS and A-level standard. Total: 135
marks. Suggested time: 2 hours 15 minutes.
Instructions
These questions are original, written in the style and difficulty of UK A-level Pure
Mathematics papers (Edexcel, AQA, OCR). They are not reproductions of real exam
papers. Attempt every question in exam conditions first, then mark yourself using the
mark scheme at the end.
Show all working. Method marks (M) are awarded for a correct approach even if the
final answer is wrong.
Give exact answers unless asked for decimals. Otherwise give answers to 3 significant
figures.
Calculators are allowed, but write down any formula you use.
The mark for each part is shown in square brackets, e.g. [3].
How to Use This Pack at Your Level
Every question is tagged Foundation, Core or Stretch, so you can start where you are and
work up.
Level Who it suits What to attempt Marks Support to use
Foundation New to A-level Q1, Q2, Q7, Q10, Q12 32 Hints, glossary,
maths, or bridging and as much time
from GCSE as you need
Core Studying AS or A- All Foundation and 111 Hints only after a
level and wanting Core questions first attempt
exam practice
Stretch Aiming for top The whole pack 135 Mark scheme only
grades or preparing timed, then the plus 16 once you have
for Further Maths extension problems finished
Move up a level when you score about 75% at your current one. At Foundation that is 24
out of 32. Core questions are Q3, Q4, Q5, Q6, Q8, Q9, Q11, Q13 and Q16. Stretch questions
are Q14, Q15, Q17 and Q18.
Page 1 of 14
, Pure Mathematics Practice Pack: Past Paper Style Questions
Contents
1. Section A: Algebra and Functions (Q1 to Q4)
2. Section B: Coordinate Geometry (Q5 to Q6)
3. Section C: Sequences, Series and the Binomial Expansion (Q7 to Q8)
4. Section D: Trigonometry (Q9 to Q10)
5. Section E: Exponentials, Logarithms and Graphs (Q11 to Q12)
6. Section F: Differentiation (Q13 to Q15)
7. Section G: Integration (Q16 to Q17)
8. Section H: Proof (Q18)
9. Extension problems, hints, mark scheme and worked solutions
10. Formula reference, plain-English glossary and self-assessment
Section A: Algebra and Functions
Question 1 [8 marks] · Foundation
f(x) = 3x² − 12x + 7
(a) Write f(x) in the form a(x + b)² + c, where a, b and c are constants. [3]
(b) State the minimum value of f(x) and the value of x at which it occurs. [2]
(c) Hence solve f(x) = 0, giving your answers as exact values. [3]
Question 2 [7 marks] · Foundation
(a) Solve the inequality 3x − 7 < 5x + 1. [2]
(b) Solve the inequality x² − x − 6 > 0. [3]
(c) Hence find the set of values of x for which both inequalities are true. [2]
Question 3 [6 marks] · Core
f(x) = 2x³ − 3x² − 11x + 6
(a) Use the factor theorem to show that (x − 3) is a factor of f(x). [2]
(b) Hence factorise f(x) completely. [3]
(c) Write down all the solutions of f(x) = 0. [1]
Question 4 [9 marks] · Core
The functions f and g are defined for all real x by
Page 2 of 14
Pure Mathematics Practice Pack
Past paper style questions with full mark scheme, AS and A-level standard. Total: 135
marks. Suggested time: 2 hours 15 minutes.
Instructions
These questions are original, written in the style and difficulty of UK A-level Pure
Mathematics papers (Edexcel, AQA, OCR). They are not reproductions of real exam
papers. Attempt every question in exam conditions first, then mark yourself using the
mark scheme at the end.
Show all working. Method marks (M) are awarded for a correct approach even if the
final answer is wrong.
Give exact answers unless asked for decimals. Otherwise give answers to 3 significant
figures.
Calculators are allowed, but write down any formula you use.
The mark for each part is shown in square brackets, e.g. [3].
How to Use This Pack at Your Level
Every question is tagged Foundation, Core or Stretch, so you can start where you are and
work up.
Level Who it suits What to attempt Marks Support to use
Foundation New to A-level Q1, Q2, Q7, Q10, Q12 32 Hints, glossary,
maths, or bridging and as much time
from GCSE as you need
Core Studying AS or A- All Foundation and 111 Hints only after a
level and wanting Core questions first attempt
exam practice
Stretch Aiming for top The whole pack 135 Mark scheme only
grades or preparing timed, then the plus 16 once you have
for Further Maths extension problems finished
Move up a level when you score about 75% at your current one. At Foundation that is 24
out of 32. Core questions are Q3, Q4, Q5, Q6, Q8, Q9, Q11, Q13 and Q16. Stretch questions
are Q14, Q15, Q17 and Q18.
Page 1 of 14
, Pure Mathematics Practice Pack: Past Paper Style Questions
Contents
1. Section A: Algebra and Functions (Q1 to Q4)
2. Section B: Coordinate Geometry (Q5 to Q6)
3. Section C: Sequences, Series and the Binomial Expansion (Q7 to Q8)
4. Section D: Trigonometry (Q9 to Q10)
5. Section E: Exponentials, Logarithms and Graphs (Q11 to Q12)
6. Section F: Differentiation (Q13 to Q15)
7. Section G: Integration (Q16 to Q17)
8. Section H: Proof (Q18)
9. Extension problems, hints, mark scheme and worked solutions
10. Formula reference, plain-English glossary and self-assessment
Section A: Algebra and Functions
Question 1 [8 marks] · Foundation
f(x) = 3x² − 12x + 7
(a) Write f(x) in the form a(x + b)² + c, where a, b and c are constants. [3]
(b) State the minimum value of f(x) and the value of x at which it occurs. [2]
(c) Hence solve f(x) = 0, giving your answers as exact values. [3]
Question 2 [7 marks] · Foundation
(a) Solve the inequality 3x − 7 < 5x + 1. [2]
(b) Solve the inequality x² − x − 6 > 0. [3]
(c) Hence find the set of values of x for which both inequalities are true. [2]
Question 3 [6 marks] · Core
f(x) = 2x³ − 3x² − 11x + 6
(a) Use the factor theorem to show that (x − 3) is a factor of f(x). [2]
(b) Hence factorise f(x) completely. [3]
(c) Write down all the solutions of f(x) = 0. [1]
Question 4 [9 marks] · Core
The functions f and g are defined for all real x by
Page 2 of 14
